Lieb and hole-doped ferrimagnetism, spiral, resonating valence-bond states, and phase separation in large-U $AB_{2}$ Hubbard chains

TL;DR

This paper investigates the ground state properties of the $AB_2$ Hubbard chain, revealing ferrimagnetic, spiral, resonating valence-bond states, and phase separation phenomena through analytical and numerical methods.

Contribution

It provides a comprehensive analytical framework for understanding charge and spin quantum fluctuations in doped $AB_2$ Hubbard chains, including phase transitions and magnetic order collapse.

Findings

Identification of spatially modulated spin structures.

Discovery of phase separation in doped regimes.

Collapse of spiral ferrimagnetic order at critical doping.

Abstract

The ground state (GS) properties of the quasi-one-dimensional $AB_2$ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. In the strong-coupling regime, we derive a low-energy Lagrangian suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and Lieb total spin per unit cell of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding $t\textrm{-}J$ Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonating-valence-bond states. We also calculate the doping-dependent GS energy and total spin per unit cell, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure.

Equations

$$\mathcal{H}=-\sum _{⟨i\alpha ,j\beta ⟩\sigma }\{t_{ij}^{\alpha \beta }{\hat{c}}_{i\alpha \sigma }^{†}{\hat{c}}_{j\beta \sigma }+\text{H.c.}\}+U\sum _{i\alpha }{\hat{n}}_{i\alpha \uparrow }{\hat{n}}_{i\alpha \downarrow },$$

$${\hat{n}}_{i\alpha \uparrow }{\hat{n}}_{i\alpha \downarrow }=\frac{1}{2}{\hat{\rho }}_{i\alpha }-2({\hat{\mathbf{S}}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }{)}^2,$$

$${\hat{\mathbf{S}}}_{i\alpha }=1/2\sum _{\sigma {\sigma }^{{}^{\prime }}}{\hat{c}}_{i\alpha {\sigma }^{{}^{\prime }}}^{†}{\mathbf{\sigma }}_{{\sigma }^{{}^{\prime }}\sigma }{\hat{c}}_{i\alpha \sigma },$$

$${\hat{\rho }}_{i\alpha }={\hat{n}}_{i\alpha \uparrow }+{\hat{n}}_{i\alpha \downarrow },$$

$$\mathcal{Z}=\int \prod _{i\alpha \sigma }\mathcal{D}{c}_{i\alpha \sigma }^{†}\mathcal{D}{c}_{i\alpha \sigma }\prod _{i\alpha }{\mathcal{D}}^2{\mathbf{n}}_{i\alpha }W(\{{\mathbf{n}}_{i\alpha }\})e^{-{\int }_0^{\beta }\mathcal{L}(\tau )d\tau },$$

$$\mathcal{D}{c}_{i\alpha \sigma }^{†}\mathcal{D}{c}_{i\alpha \sigma }\equiv \underset{M\to \infty ,\delta \tau \to 0}{\lim }\prod _{r=1}^{M-1}d{c}_{i\alpha \sigma }^{†}({\tau }_r)d{c}_{i\alpha \sigma }({\tau }_r),$$

$${\mathcal{D}}^2{\mathbf{n}}_{i\alpha }\equiv \underset{M\to \infty ,\delta \tau \to 0}{\lim }\prod _{r=1}^{M-1}{d}^2{\mathbf{n}}_{i\alpha }({\tau }_r),$$

$$\int \prod _{i\alpha }{d}^2{\mathbf{n}}_{i\alpha }W(\{{\mathbf{n}}_{i\alpha }({\tau }_r)\})=1,$$

$$\mathcal{L}(\tau )$$

$$+U\sum _{i\alpha }[\frac{{\rho }_{i\alpha }}{2}-2({\mathbf{S}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }{)}^2].$$

$$2({\hat{\mathbf{S}}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }{)}^2=\frac{{\hat{\rho }}_{i\alpha }(2-{\hat{\rho }}_{i\alpha })}{2},$$

$$p_{i\alpha }{\hat{\mathbf{S}}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }=\frac{{\hat{\rho }}_{i\alpha }(2-{\hat{\rho }}_{i\alpha })}{2},$$

$$W$$

$$\displaystyle=\mathcal{C}\exp\{-\int_{0}^{\beta}d\tau\gamma\sum_{i\alpha}[p_{i\alpha}\mathbf{S}_{i\alpha}\cdot\mathbf{n}_{i\alpha}-\frac{\rho_{i\alpha}}{2}(2-\rho_{i\alpha})]^{2}\Big{\}},$$

$$\mathcal{L}(\tau )$$

$$+U\sum _{i\alpha }[\frac{{\rho }_{i\alpha }}{2}-p_{i\alpha }({\mathbf{S}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha })],$$

$$U_{i\alpha}=\left[\begin{array}[]{cc}\cos\left(\frac{{\textstyle\theta_{i\alpha}}}{{\textstyle 2}}\right)&-\sin\left(\frac{{\textstyle\theta_{i\alpha}}}{{\textstyle 2}}\right)e^{-i\phi_{i\alpha}}\\ \sin\left(\frac{{\textstyle\theta_{i\alpha}}}{{\textstyle 2}}\right)e^{i\phi_{i\alpha}}&\cos\left(\frac{{\textstyle\theta_{i\alpha}}}{{\textstyle 2}}\right)\end{array}\right],$$

$$c_{i\alpha \sigma }=\sum _{{\sigma }^{\prime }}(U_{i\alpha }{)}_{\sigma {\sigma }^{\prime }}{a}_{i\alpha {\sigma }^{\prime }},$$

$$\boldsymbol{\sigma}\cdot\mathbf{n}_{i\alpha}=\left[\begin{array}[]{cc}\cos\left(\theta_{i\alpha}\right)&\sin\left(\theta_{i\alpha}\right)e^{-i\phi_{i\alpha}}\\ \sin\left(\theta_{i\alpha}\right)e^{i\phi_{i\alpha}}&-\cos\left(\theta_{i\alpha}\right)\end{array}\right],$$

$$U_{i\alpha }^{†}(\mathbf{\sigma }\cdot {\mathbf{n}}_{i\alpha })U_{i\alpha }={\sigma }^z,$$

$${\mathbf{S}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }$$

$$=\frac{1}{2}\sum _{\sigma {\sigma }^{{}^{\prime }}}{a}_{i\alpha \sigma }^{†}({\sigma }_z{)}_{\sigma {\sigma }^{{}^{\prime }}}{a}_{i\alpha {\sigma }^{{}^{\prime }}}\equiv S_{i\alpha }^z;$$

$${\mathbf{S}}_{i\alpha }\cdot {\mathbf{n}}_{i\alpha }=p_{i\alpha }\frac{{\rho }_{i\alpha }(2-{\rho }_{i\alpha })}{2}=\frac{1}{2}(a_{i\alpha \uparrow }^{†}{a}_{i\alpha \uparrow }-a_{i\alpha \downarrow }^{†}{a}_{i\alpha \downarrow }),$$

$$\mathcal{L}(\tau )={\mathcal{L}}_0(\tau )+{\mathcal{L}}_n(\tau ),$$

$${\mathcal{L}}_0(\tau )=$$

$$+\frac{U}{2}\sum _{i\alpha \sigma }(1-p_{i\alpha }\sigma )a_{i\alpha \sigma }^{†}{a}_{i\alpha \sigma },$$

$${\mathcal{L}}_n$$

$$-\sum _{i\alpha j\beta \sigma {\sigma }^{\prime }}{t}_{ij}^{\alpha \beta }[a_{i\alpha {\sigma }^{\prime }}^{†}(U_{i\alpha }^{†}{U}_{j\beta }-1{)}_{{\sigma }^{\prime }\sigma }{a}_{j\beta \sigma }+\text{H.c.}],$$

$$\delta =1-\frac{1}{N}\sum _{i\alpha }⟨{\rho }_{i\alpha }⟩,$$

$${\mathcal{H}}_0=$$

$$+\frac{U}{2}\sum _{i\alpha \sigma }(1-p_{i\alpha }\sigma )a_{i\alpha \sigma }^{†}{a}_{i\alpha \sigma }.$$

$$(d_{i\sigma },e_{i\sigma })=\frac{1}{\sqrt{2}}(a_{i{B}_1\sigma }\pm a_{i{B}_2\sigma }),b_{i\sigma }=a_{iA\sigma }.$$

$${\mathcal{H}}_0=\sum _{k\sigma }{\epsilon }_k[A_{k\sigma }^{†}{A}_{k\sigma }-B_{k\sigma }^{†}{B}_{k\sigma }]+\frac{U}{2}\sum _{k\sigma }(1-\sigma )e_{k\sigma }^{†}{e}_{k\sigma }$$

$$+\frac{U}{2}\sum _{k\sigma }[A_{k\sigma }^{†}{A}_{k\sigma }+B_{k\sigma }^{†}{B}_{k\sigma }-\sigma (A_{k\sigma }^{†}{B}_{k\sigma }+B_{k\sigma }^{†}{A}_{k\sigma })],$$

Full text

I introduction

Much attention has been given to quantum phase transitions Sachdev (2011); Continentino (2017), which are phenomena characterized by the change of the nature of the ground state (GS) driven by a non-thermal parameter: pressure, magnetic field, doping, Coulomb repulsion, or competitive interactions. In this context, the study of quasi-one-dimensional (quasi-1D) compounds with ferrimagnetic properties Coutinho-Filho et al. (2008); Ivanov (2009) has attracted considerable theoretical and experimental interest because of their unique physical properties and very rich phase diagrams. In particular, the GS of quasi-1D quantum ferrimagnets with $AB_{2}$ or $ABB^{\prime}$ unit cell topologies (diamond or trimer chains) described by the Heisenberg or Hubbard models Montenegro-Filho and Coutinho-Filho (2005) exhibit unsaturated spontaneous magnetization, ferromagnetic and antiferromagnetic spin-wave modes, effect of quantum fluctuations, and field-dependent magnetization plateaus, among several other features of interest.

Of special interest is the topological origin of GS magnetic long-range order associated with the unit cell structure of the lattice Montenegro-Filho and Coutinho-Filho (2005); Silvestre and Hoffmann (1985); Macêdo et al. (1995); Tian and Lin (1996); Alcaraz and Malvezzi (1997); Raposo and Coutinho-Filho (1997); *Raposo1999; Sierra et al. (1999); Martín-Delgado et al. (2005). These studies have been motivated and supported by exact solutions and rigorous results Lieb and Mattis (1962); Lieb and Wu (1968); Lieb (1989); Lie (1995); Tasaki (1998a); *Tasaki1998nagaoka; Tian (2004); in particular, at half filling, the total spin per unit cell obeys Lieb-Mattis Lieb and Mattis (1962) (Heisenberg model) or Lieb’s theorem Lieb (1989) (Hubbard model). On the other hand, it has been verified that the ferrimagnetic GS of spin-$1/2$ Heisenberg and Hubbard/$$t-$J$ $AB_{2}$ chains, under the effect of frustration Yamamoto and Ohara (2007); Montenegro-Filho and Coutinho-Filho (2008); Hida and Takano (2008); Shimokawa and Nakano (2012); Furuya and Giamarchi (2014) or doping Macêdo et al. (1995); Sierra et al. (1999); Montenegro-Filho and Coutinho-Filho (2006, 2014), are strongly affected by quantum fluctuations that might cause its destruction and the occurrence of new exotic phases: spiral incommensurate (IC) spin structures, Nagaoka ($U\rightarrow\infty$) and resonating-valence-bond (RVB) states, phase separation (PS), and Luttinger-liquid behavior. These features can enhance the phenomenology in comparison with a linear chain, which is dominated by the nontrivial Luttinger-liquid behavior that exhibits fractional excitations Giamarchi (2004); Essler et al. (2005), emergent fractionalized particles Carmelo and Sacramento (2018), and fractional-exclusion statistic properties Haldane (1991); *Wu_1994 in the spin-incoherent regime Vitoriano et al. (2018). In addition, investigations of transport properties in $AB_{2}$ chains, and related structures, have also unveiled very interesting features Lopes and Dias (2011); *Lopes2014.

On the experimental side, studies Drillon et al. (1993); Belik et al. (2005); Matsuda et al. (2005) of the magnetic properties of homometallic phosphate compounds of the family $\mathrm{A_{3}Cu_{3}\left(PO_{4}\right)_{4}}$ ($\mathrm{A=Ca}$, $\mathrm{Sr}$, $\mathrm{Pb}$) suggest that in these materials the line of trimers formed by spin-$1/2$ $\mathrm{Cu^{+2}}$ ions antiferromagnetically coupled do exhibit ferrimagnetism of topological origin. Further, compounds $\mathrm{Ca_{3}}\mathrm{M}_{3}(\mathrm{PO_{4}})_{4}(\mathrm{M}=\mathrm{Ni},\mathrm{Co})$ with a wave-like layer structure built by zigzag M-chains exhibit antiferromagnetic ordering ($\mathrm{M}=\mathrm{Ni}$) or paramagnetic behavior ($\mathrm{M}=\mathrm{Co}$) Guo et al. (2017). On the other hand, bimetallic compounds, such as $\mathrm{CuMn\left(S_{2}C_{2}O_{2}\right)_{2}\cdot 7.5H_{2}O}$ Verdaguer et al. (1984), can be modeled Verdaguer et al. (1984); Tenório et al. (2011); Strečka and Verkholyak (2016); *[Seealso:][]Yan2015 by alternate spin-$1/2$ - spin-$5/2$ chains and support interesting field-induced quantum critical points and Luttinger-liquid phase Tenório et al. (2011). In addition, frustrated diamond ($AB_{2}$ topology) chains can properly model the compound azurite, $\mathrm{Cu_{3}(CO_{3})_{2}(OH)_{2}}$, in which case the occurrence of the $1/3$ magnetization plateau is verified at high fields Kikuchi et al. (2005) in agreement with topological arguments Oshikawa et al. (1997) akin to those invoked in the quantum Hall effect. The spin-1/2 trimer chain compound $\mathrm{Cu_{3}(P_{2}O_{6}OH)_{2}}$, with antiferromagnetic interactions only, also display the 1/3 magnetization plateau Hase et al. (2006). Interestingly, it has been established that in azurite the magnetization plateau is a dimer-monomer state Rule et al. (2008), i.e., the chain is formed by pairs of $S=1/2$ monomers and $S=0$ dimers, with a small local polarization of the diamond spins Aimo et al. (2009), in agreement with density functional theory Jeschke et al. (2011). These dimer-monomer states have been found previously in the context of modeling frustrated $AB_{2}$ chains Takano et al. (1996); Okamoto et al. (1999, 2003), and confirmed through a modeling using quantum rotors Tenório et al. (2009). In contrast to azurite, whose dimers appear perpendicular to the chain direction, in the spin-1/2 inequilateral diamond-chain compounds Morita et al. (2017) $\mathrm{A}_{3}\mathrm{Cu}_{3}\mathrm{AlO}_{2}(\mathrm{SO}_{4})_{4}(\mathrm{A}=\mathrm{K},\mathrm{Rb},\mathrm{Cs})$, the magnetic exchange interactions force the dimers to lie along the sides of the diamond cells and the monomers form a 1D Heisenberg chain. In fact, the low-energy excitations of these new compounds have been probed and a Tomonaga-Luttinger spin liquid behavior identified Fujihala et al. (2017). It is worth mentioning that strongly frustrated $AB_{2}$ chains can exhibit ladder-chain decoupling Montenegro-Filho and Coutinho-Filho (2008), in which case the ladder is formed via the coupling between dimer spins in neighboring $AB_{2}$ unit cells.

On the other hand, besides the above-mentioned quasi-1D compounds and related magnetic properties, considerable efforts have been devoted to the study of superconductivity and intriguing magnetic/charge ordered phases in doped materials Dagotto (2005); Han et al. (2015), in particular the formation of spin-gapped states in compounds such as the family of doped $\mathrm{(La,Sr,Ca)_{14}Cu_{24}O_{41}}$. This compound is formed by one-dimensional $\mathrm{CuO_{2}}$ diamond chains, $\mathrm{(Sr,Ca)}$ layers, and two-leg $\mathrm{Cu_{2}O_{3}}$ ladders Dagotto (1999). These results certainly stimulate experimental and theoretical investigations of quasi-1D compounds in the hole-doped regime, which is the main focus of our work, as described in the following.

In this work, we shall employ an analytical approach suitable to describe the strongly coupled Hubbard model on doped $AB_{2}$ chains, which were the object of recent numerical studies through density matrix renormalization group (DMRG) techniques Montenegro-Filho and Coutinho-Filho (2014). Our functional integral approach, combined with a perturbative expansion in the strong-coupling regime, was originally proposed to study the doped Hubbard chain Weng et al. (1991); *Weng_1992prb, and later adapted to describe various doped-induced phase transitions in the $U=\infty$ $AB_{2}$ Hubbard chain Oliveira et al. (2009). In addition, this approach was used to describe the doped strongly coupled Hubbard model on the honeycomb lattice Ribeiro and Coutinho-Filho (2015), whose results are very rewarding, particularly those for the GS energy and magnetization in the doped regime, which compare very well with Grassmann tensor product numerical studies Gu et al. (2013).

The paper is organized as follows: in Sec. II we review the functional integral representation of the Hubbard Hamiltonian in terms of Grassmann fields (charge degrees of freedom) and spin $SU(2)$ gauge fields (spin degrees of freedom). In Sec. III we diagonalize the Hamiltonian associated with the charge degree of freedom and obtain a perturbative low-energy theory suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regimes. In Sec. IV, we show that the resultant Hamiltonian at half filling and large-U maps onto the spin-$1/2$ quantum Heisenberg model. In this regime, a perturbative series expansions in powers of $1/S$ of the spin-wave modes is presented, which allows us to calculate the GS energy, sublattice magnetizations, and Lieb GS total spin per unit cell in very good agreement with previous estimates. In Sec. V, we derive the low-energy effective $t$-$J$ Hamiltonian, which accounts for both charge and spin quantum fluctuations. We also present the evolution of the second-order spin-wave modes, GS energy and total spin per unit cell under hole doping, thus identifying the occurrence of spatially modulated spin structures, with non-zero and zero GS total spin, and phase separation involving the later spin structure and RVB states at hole concentration $1/3$. Remarkably, these predictions are in very good agreement with the DMRG data reported in Ref. [Montenegro-Filho and Coutinho-Filho, 2014]. Lastly, in Sec. VI, we present a summary and concluding remarks concerning the reported results.

II Functional-Integral representation

The Hamiltonian of the one-band Hubbard model on chains with $AB_{2}$ unit cell topology is given by Macêdo et al. (1995); Tian and Lin (1996); Raposo and Coutinho-Filho (1997, 1999):

[TABLE]

where $i=1,\ldots,N_{c}\,(=N/3)$ is the specific position of the unit cell, whose length is set to unity, $N_{c}$ ($N$) is the number of cells (sites), $\alpha,\beta=A,\,B_{1},\,B_{2}$ denote the type of site within the unit cell, $\hat{c}_{i\alpha\sigma}^{\dagger}$ ($\hat{c}_{i\alpha\sigma}$) is the creation (annihilation) operator of electrons with spin $\sigma$ ($=\uparrow,\downarrow$) at site $\alpha$ of cell $i$, and $\hat{n}_{i\alpha\sigma}=\hat{c}_{i\alpha\sigma}^{\dagger}\hat{c}_{i\alpha\sigma}$ is the occupancy number operator. The first term in Eq. (1) describes electron hopping, with energy $t_{ij}^{\alpha\beta}\equiv t$, allowed only between nearest neighbors $A\textrm{-}B_{1}$ and $A\textrm{-}B_{2}$ linked sites of sublattices $A$ and $B$ (bipartite lattice), and the second one is the on-site Coulombian repulsive interaction $U>0$, which contributes only in the case of double occupancy of the site $i\alpha$.

At this point, it is instructive to digress on some fundamental aspects of the formalism used in our work Weng et al. (1991, 1992); Oliveira et al. (2009); Ribeiro and Coutinho-Filho (2015). With regard to the large-U doped Hubbard chain Weng et al. (1991, 1992), $U=\infty$ $AB_{2}$ Hubbard chain Oliveira et al. (2009) and the Hubbard model on the honeycomb lattice Ribeiro and Coutinho-Filho (2015), it has been shown that the particle density product in Eq. (1) can be treated through the use of a decomposition procedure, which consists in expressing $\hat{n}_{i\alpha\uparrow}\hat{n}_{i\alpha\downarrow}$ in terms of charge and spin operators:

[TABLE]

where

[TABLE]

and

[TABLE]

are the spin-1/2 and charge-density operators, respectively, $\mathbf{\sigma}_{\sigma^{{}^{\prime}}\sigma}$ denotes the Pauli matrix elements $(\hbar\equiv 1)$, and $\mathbf{n}_{i\alpha}$ is an arbitrary unit vector. In fact, Eq. (2) follows from the identity: $\frac{1}{2}\hat{\rho}_{i\alpha}-\hat{n}_{i\alpha\uparrow}\hat{n}_{i\alpha\downarrow}=2(\hat{S}_{i\alpha}^{x,y,z})^{2}=2(\hat{\mathbf{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})^{2}$. The convenience of using the decomposition defined in Eq. (2), with explicit spin-rotational invariance for the large-U Hubbard model, was discussed at length in Refs. [Weng et al., 1991, 1992; Oliveira et al., 2009; Ribeiro and Coutinho-Filho, 2015].

We start by using the Trotter-Suzuki formula Fradkin (2013); Negele and Orland (1988), which allows us to write the partition function, $\mathcal{Z}=\textrm{Tr}\left[\exp(-\beta\mathcal{H})\right]$, at a temperature $k_{B}T\equiv 1/\beta$, as $\mathcal{Z}=\textrm{Tr}\{\hat{T}\prod_{r=1}^{M}\exp[-\delta\tau\mathcal{H}(\tau_{r})]\}$, where $\hat{T}$ denotes the time-ordering operator, the total imaginary time interval is formally sliced into $M$ discrete intervals of equal size $\delta\tau=\tau_{r}-\tau_{r-1}$, $r=1,2,...,M,$ with $\tau_{0}=0,$ and $\tau_{M}=\beta=M\delta\tau$, under the limits $M\rightarrow\infty$ and $\delta\tau\rightarrow 0$. We shall now introduce, between each discrete time interval, an overcomplete basis of fermionic coherent states Fradkin (2013); Negele and Orland (1988), $1=\int\prod_{i\alpha\sigma}dc_{i\alpha\sigma}^{\dagger}dc_{i\alpha\sigma}\exp(-\sum_{i\alpha\sigma}c_{i\alpha\sigma}^{\dagger}c_{i\alpha\sigma})\arrowvert\{c_{i\alpha\sigma}\}\rangle\langle\{c_{i\alpha\sigma}\}\arrowvert$, where $\{c_{i\alpha\sigma}^{\dagger},c_{i\alpha\sigma}$} denotes a set of Grassmann fields satisfying anti-periodic boundary conditions: $c_{i\alpha\sigma}^{\dagger}(0)=-c_{i\alpha\sigma}^{\dagger}(\beta)$ and $c_{i\alpha\sigma}(0)=-c_{i\alpha\sigma}(\beta)$; while the set of unit vectors defines the vector field $\{\mathbf{n}_{i\alpha}\}$, satisfying periodic ones: $\mathbf{n}_{i\alpha}(0)=\mathbf{n}_{i\alpha}(\beta)$, under a weight functional (see below). Thereby, following standard procedure Fradkin (2013); Negele and Orland (1988), the partition function reads:

[TABLE]

where the pertinent measures are defined by

[TABLE]

[TABLE]

the weight functional, $W(\{\mathbf{n}_{i\alpha}\})$, satisfies a normalization condition at each discrete imaginary time $\tau_{r}$:

[TABLE]

and the Lagrangian density $\mathcal{L}(\tau)$ is written in the form:

[TABLE]

In order to fix $W(\{\mathbf{n}_{i\alpha}\})$ one should notice that, in the operator formalism: $\hat{\rho}_{i\alpha}^{2}=\hat{\rho}_{i\alpha}+2\hat{n}_{i\alpha\uparrow}\hat{n}_{i\alpha\downarrow}$. Therefore, using Eq. (2), the following identity holds Weng et al. (1991, 1992):

[TABLE]

which means that the square of the spin component operator along the $\mathbf{n}_{i\alpha}$ direction has zero eigenvalues if the site is vacant or doubly occupied, and a nonzero value only for singly occupied sites, i.e., ${(\mathbf{\hat{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})^{2}=1/4}$. Now, taking advantage of the choice of $\mathbf{n}_{i\alpha}$, the local spin-polarization and spin-quantization axes are both chosen along the $\mathbf{n}_{i\alpha}$ direction. Therefore, for singly occupied sites, we find $\mathbf{S}_{i\alpha}\cdot\mathbf{n}_{i\alpha}=p_{i\alpha}/2$, with $p_{i\alpha}=\pm 1$, corresponding to the two possible spin-1/2 states. Further, by incorporating vacancy and double occupancy possibilities, corresponding to the four possible local states of the Hubbard model, one can write Weng et al. (1991, 1992)

[TABLE]

with $p_{i\alpha}^{2}=(\pm 1)^{2}$. We stress that, due to fermion operator properties, the square of Eq. (11) reproduces Eq. (10), and a comparison between them implies, at arbitrary doping and U value, the formal equivalence between $2(\hat{\mathbf{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})^{2}$ and $p_{i\alpha}(\hat{\mathbf{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})$. In this context, we remark that the original Coulomb repulsion term of the Hubbard Hamiltonian in Eq. (1) is formally and energetically (eigenvalues) equivalent to both that in Eq. (9) or in its linear version through the following replacement: $2(\hat{\mathbf{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})^{2}\rightarrow p_{i\alpha}(\hat{\mathbf{S}}_{i\alpha}\cdot\mathbf{n}_{i\alpha})$. Indeed, using the constraint in Eq. (11) we find, $U\sum_{i\alpha}[\frac{\rho_{i\alpha}}{2}-p_{i\alpha}(\mathbf{S}_{i\alpha}\cdot\mathbf{n}_{i\alpha})]=U\sum_{i\alpha}[\frac{\rho_{i\alpha}}{2}-\frac{1}{2}\rho_{i\alpha}(2-\rho_{i\alpha})]$, which is zero for $\rho_{i\alpha}=0,1$; whereas, as expected, for double occupied sites, $\rho_{i\alpha}=2$, the local energy is $U$. Therefore, Eq. (11) in its Grassmann version, can be enforced by a proper choice of the normalized weight functional:

[TABLE]

where $\gamma\rightarrow\infty$ in the continuum limit ($M\rightarrow\infty$, $\delta\tau\rightarrow 0$), with delta-function peaks at the four local states of the Hubbard model, and $\mathcal{C}$ is a normalization factor such that Eq. (8) holds. In fact, the product of $W(\{\mathbf{n}_{i\alpha}(\tau_{r})\})$ in Eq. (12) generates a sum in $r$ in the exponential of the suitable chosen Gaussian function, i.e., $W(\{\mathbf{n}_{i\alpha}\})$ is such that in the continuum limit, $M\rightarrow\infty,\delta\tau\rightarrow 0$, Eq. (12) obtains with a diverging $\gamma$, as pointed out in Ref. [Weng et al., 1992]. In this way, using Eq. (12) for the weight functional in Eq. (5) for the partition function $\mathcal{Z}$, and integrating over $\{\mathbf{n}_{i\alpha}\}$, the Lagrangian density $\mathcal{L}(\tau)$ in Eq. (9) can thus be written in the following linearized form Weng et al. (1991):

[TABLE]

where the constraint in Eq. (11) was explicitly used.

Now, since we are interested in studying the GS properties of the $AB_{2}$ Hubbard chains, we choose the staggered factor $p_{i\alpha}=+1\,\,(-1)$ at sites $\alpha=B_{1},\,B_{2}$ $(A)$, consistent with the long-range ferrimagnetic GS predicted by Lieb’s theorem at half filling and for any $U$ value Lieb (1989); Macêdo et al. (1995); Tian and Lin (1996), in which case we assume broken rotational symmetry along the $z$-axis. In this context, by considering the symmetry exhibited by the ferrimagnetic order, let us define the $SU(2)/U(1)$ unitary rotation matrix Tung (1985)

[TABLE]

where $\theta_{i\alpha}$ is the polar angle between the $z$-axis and the unit local vector $\mathbf{n}_{i\alpha}$ and $\phi_{i\alpha}\in[0,2\pi)$ is an arbitrary azimuth angle due to the $U(1)$ gauge freedom of choice for $U_{i\alpha}$. Moreover, a new set of Grassmann fields, {$a_{i\alpha\sigma}^{\dagger},a_{i\alpha\sigma}$} can be obtained, according to the transformation:

[TABLE]

that locally rotates each unit vector $\mathbf{n}_{i\alpha}$ to the $z$-direction. On the other hand, if we express the product $\boldsymbol{\sigma}\cdot\mathbf{n}_{i\alpha}$ in matrix form:

[TABLE]

we obtain, after using Eq. (14),

[TABLE]

which explicitly manifest the broken rotational symmetry along the $z$-axis. In this way, by substituting Eqs. (14) and (15) into Eq. (3), and using the above result, we find

[TABLE]

thereby, the constraint in Eq. (11) can be written in the form

[TABLE]

where $p_{i\alpha}=+1\,\,(-1)$ at sites $\alpha=B_{1},\,B_{2}$ $(A)$. The choice of $p_{i\alpha}$ above implies Lieb’s ferrimagnetic ordering with the set $\{\theta_{iA}=\theta_{iB_{1}}=\theta_{iB_{2}}=0\}$, for all $i$, at half filling. However, in the hole doped regime away from half filling, the $\theta_{i\alpha}$’s can be nonzero (e.g., $\theta_{i\alpha}=\pi$ for a spin flip, leading to a change in the sign of $S_{i\alpha}^{z}$); further, $S_{i\alpha}^{z}$ can be zero either by the presence of holes or doubly occupied sites ($a_{i\alpha\uparrow}^{\dagger}a_{i\alpha\uparrow}=a_{i\alpha\downarrow}^{\dagger}a_{i\alpha\downarrow}$). Lastly, using Eqs. (15) and (19) into the Lagrangian, Eq. (13), we find, after suitable rearrangement of terms,

[TABLE]

where both Lagrangians are quadratic in the Grassmann fields:

[TABLE]

and

[TABLE]

with the first term in both Eqs. (21) and (22) being originated from the first term in Eq. (13), the second ones come from the hopping term in Eq. (13), after a rearrangement of terms, while the last one in Eq. (21) (proportional to $U$) is obtained by using Eq. (19) in the last term of Eq. (13). It is worth mentioning that only charge degrees of freedom (Grassmann fields) appear in $\mathcal{L}_{0}(\tau)$, and spin degrees of freedom under the constraint in Eq. (19) [$SU(2)$ gauge fields $\{U_{i\alpha}^{\dagger},U_{i\alpha}\}$, which carry all the information on the vector field $\{\mathbf{n}_{i\alpha}\}$] are now restricted to $\mathcal{L}_{n}(\tau)$, which includes both spin and charge degrees of freedom.

In the large-U regime, double occupancy is energetically unfavorable and the factor $2-\rho_{i\alpha}$ is no longer needed in Eq. (19), i.e., $\mathbf{S}_{i\alpha}\cdot\mathbf{n}_{i\alpha}=p_{i\alpha}\frac{\rho_{i\alpha}}{2}$, with $\rho_{i\alpha}=0$ or $1$. In this case, a proper perturbative analysis will allow us to study hole doping effects in Sec. V in a macroscopic fashion, so we define

[TABLE]

which measures the thermodynamic average of hole doping away from half filling. In this context (strong-coupling limit), we take advantage of results derived from $\mathcal{L}_{0}(\tau)$ (charge effects in Sec. III), and at half filling (Sec. IV), in which case charge degrees of freedom are frozen.

III Charge degrees of freedom and the strong-coupling limit

In this section, we shall first diagonalize the Hamiltonian associated with the Lagrangian $\mathcal{L}_{0}(\tau)$ through the use of a special symmetry property of the $AB_{2}$ chains and a canonical transformation in reciprocal space. Then, by introducing a perturbative expansion in the strong-coupling regime, a low-energy effective Lagrangian for the $AB_{2}$ Hubbard chains at half filling and in the doped regime will be obtained.

III.1 Charge degrees of freedom

We begin our discussion by considering the Lagrangian $\mathcal{L}_{0}$ in Eq. (21), and its corresponding Hamiltonian $\mathcal{H}_{0}$, free of the $SU(2)$ gauge fields. By performing the Legendre transformation: $\mathcal{H}_{0}=-\sum_{i\alpha\sigma}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{\tau}a_{i\alpha\sigma})}\partial_{\tau}a_{i\alpha\sigma}+\mathcal{L}_{0}$, where $\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{\tau}a_{i\alpha\sigma})}=a_{i\alpha\sigma}^{\dagger}$, the resulting $\mathcal{H}_{0}$ is given by

[TABLE]

Further, since $\mathcal{H}_{0}$ ($\mathcal{L}_{0}$) is quadratic in the Grassmann fields, the solution for the energy of the system is given by $\mathcal{H}_{0}$ in its diagonalized form Negele and Orland (1988).

The $AB_{2}$ unit cell topology exhibits a symmetry Alcaraz and Malvezzi (1997); Sierra et al. (1999); Montenegro-Filho and Coutinho-Filho (2006); Oliveira et al. (2009); Montenegro-Filho and Coutinho-Filho (2014) under the exchange of the labels of the $B$ sites in a given unit cell. Thus, we can construct a new set of Grassmann fields possessing this symmetry, i.e., either symmetric or antisymmetric with respect to the exchange operation $B_{1}\leftrightarrow B_{2}$:

[TABLE]

In addition, as a signature of the quasi-1D structure of the $AB_{2}$ chains, we notice that the $B_{1}$ and $B_{2}$ sites are located at a distance $1/2$ (in units of length) ahead of the $A$ site. Therefore, after Fourier transforming the above Grassmann fields, i.e., $\{d_{i,\sigma},e_{i,\sigma},b_{i,\sigma}\}=\frac{1}{\sqrt{N_{c}}}\sum_{k}e^{ikx_{i}}\{d_{k,\sigma},e_{k,\sigma},b_{k,\sigma}\}$, it is convenient to introduce a phase factor $e^{\frac{ik}{2}}$ through the following transformation Oliveira et al. (2009): $(A_{k\sigma},B_{k\sigma})=\frac{1}{\sqrt{2}}(d_{k\sigma}\pm e^{\frac{ik}{2}}b_{k\sigma})$, so that $\mathcal{H}_{0}$ in Eq. (24) thus becomes

[TABLE]

where

[TABLE]

with $k=2\pi j(\frac{3}{N})-\pi$, and $j=1,\ldots,N/3$. We can now exactly diagonalize $\mathcal{H}_{0}$ through the following Bogoliubov transformation:

[TABLE]

with $u_{k}$ and $v_{k}$ satisfying the canonical constraint: $(u_{k})^{2}+(v_{k})^{2}=1$, to maintain the anticommutation relations of the Grassmann fields. Due to the ferrimagnetic order of the GS, the above transformation is subject to a $4\pi$ periodicity of the Bogoliubov functions $\left\{u_{k},v_{k}\right\}$ and Grassmann fields $\left\{\alpha_{k\sigma},\beta_{k\sigma}\right\}$. The diagonalized $\mathcal{H}_{0}$ thus reads:

[TABLE]

where

[TABLE]

and

[TABLE]

As one can see from Eq. (29), the non-interacting tight binding ($U=0$) spectrum of $\mathcal{H}_{0}$ present three electronic bands: a nondispersive flat band (related to the Grassmann fields $\{e_{k\sigma}^{\dagger},e_{k\sigma}\}$, macroscopically degenerate), and two dispersive ones. In $AB_{2}$ chains, flat bands are closely associated with ferrimagnetism (unsaturated ferromagnetism) Macêdo et al. (1995); Tian and Lin (1996); Montenegro-Filho and Coutinho-Filho (2005) at half filling, in agreement with Lieb’s theorem Lieb (1989); Lie (1995), or fully polarized ferromagnetism Tasaki (1998b) associated with the flat lowest band. We also stress that even at this level of approximation and in the weak coupling regime ($U=2t$), it was shown Macêdo et al. (1995) that hole doping [parametrized by $\delta$ defined in Eq. (23)] can destroy the ferrimagnetic order and/or induce phase separation in $AB_{2}$ chains. As depicted in Fig. 1LABEL:sub@Fig_1a, the $U=0$ spin degeneracy of the flat bands is removed by the Coulombian repulsive interaction, in which case a gap $U$ opens between the $e_{k\sigma}$ modes: $e_{k\uparrow}=0$, where spins at sites $B_{1}$ and $B_{2}$ are up, and $e_{k\downarrow}=U$, where these spins are down. On the other hand, the two dispersive bands are spin degenerated, and also display a Hubbard gap $U$ separating the low $(\alpha_{k\sigma})$-energy and high $(\beta_{k\sigma})$-energy modes Oliveira et al. (2009).

III.2 Strong-coupling limit

In this subsection, we shall introduce a perturbative expansion in the strong-coupling regime ($U\gg t$) in order to obtain a low-energy effective Lagrangian for the $AB_{2}$ Hubbard chain at half filling and in the doped regime. First, we resume the results of the previous section by writing the Grassmann fields $d_{i\sigma}$ and $b_{i\sigma}$ in terms of the Grassmann (Bogoliubov) fields $\alpha_{k\sigma}$ and $\beta_{k\sigma}$:

[TABLE]

where the phase factor $e^{-\frac{ik}{2}}$ signalizes the quasi-1D $AB_{2}$ structure, and the antisymmetric Grassmann field $e_{i,\sigma}$ remains as defined in Eq. (25). In the strong-coupling limit, however, it will prove useful to define a set of auxiliary spinless Grassmann fields Weng et al. (1991, 1992); Oliveira et al. (2009) in direct space associated with $d_{i\sigma}$ and $b_{i\sigma}$:

[TABLE]

and a similar equation for ${(\alpha_{i}^{\frac{1}{2}},\beta_{i}^{\frac{1}{2}})}\leftrightarrow(\alpha_{k\sigma},\beta_{k\sigma})$ is obtained by the replacements: $\theta(\pm\sigma)\rightarrow\theta(\mp\sigma)$ and $x_{i}\rightarrow x_{i}-1/2$, where $\theta(\sigma)$ is the Heaviside function, while for the antisymmetric component, one has

[TABLE]

Now, by expanding $(u_{k},v_{k})$ in Eq. (30) in powers of $t/U$:

[TABLE]

substituting these results into the Eq. (32), and using the inverse transformation of Eq. (33), we can derive a perturbative expansion in powers of $t/U$ for the Grassmann fields $d_{i\sigma}$ and $b_{i\sigma}$ in terms of the spinless Grassmann fields as follows:

[TABLE]

[TABLE]

In the above derivation, we have used that $\theta(\sigma)\theta(\sigma^{\prime})=\theta(\sigma)\delta_{\sigma,\sigma^{\prime}}$. Notice that, since $\frac{t}{U}\ll 1$, in Eqs. (36) and (37) we can identify the fields $\alpha_{i}^{\frac{1}{2}}\approx a_{iA\downarrow}$ and $\alpha_{i}\approx(a_{iB_{1}\uparrow}+a_{iB_{2}\uparrow})/\sqrt{2}$, a result fully consistent with the low-energy spin configuration of the ferrimagnetic state discussed previously. Analogously, for the high-energy bands, the opposite spin configuration is observed, with spin up (down) present at sites $A$ ($B_{1}$, $B_{2}$).

Introducing Eqs. (36) and (37) into Eq. (24), with the aid of Eq. (25), we obtain a perturbative expression for $\mathcal{H}_{0}$ (low-energy sector) in terms of the spinless Grassmann fields up to order $J=4t^{2}/U$:

[TABLE]

By applying Fourier transform to the above expression and rearranging the terms, we obtain

[TABLE]

In Fig. 1 we plot the electronic spectrum of the Hamiltonian $\mathcal{H}_{0}$, both in the weak and strong-coupling regime: (a) Eq. (29) for $U=2t$ and (b) Eqs. (29) and (39) for $U=12t$ ($J=4t^{2}/U=1/3$), respectively, with $t\equiv 1$. We can notice the presence of the shrinking phenomenon Macêdo et al. (1995) as $U$ increases from $2t$ to $12t$ (strong-coupling regime) and that, for $U=12t$, Eq. (39) is a very good approximation to Eq. (29). Noticeably, the $t\ll U$ expansion of the fields allow us to identify $\alpha_{k}^{\frac{1}{2}}\approx a_{kA\downarrow}$, $\alpha_{k}\approx(a_{kB_{1}\uparrow}+a_{kB_{2}\uparrow})/\sqrt{2}$ (triplet state) and $e_{k\uparrow}\approx(a_{kB_{1}\uparrow}-a_{kB_{2}\uparrow})/\sqrt{2}$ (singlet state), as the low-energy spin configuration of the ferrimagnetic state with single occupancy, where spins at sites $A$($B_{1},B_{2}$) are down (up), in agreement with Lieb’s theorem Lieb (1989); Macêdo et al. (1995); Tian and Lin (1996).

In order to describe the most relevant low-energy processes that take place in this regime, one has to additionally project out the high-energy bands from $\mathcal{H}_{0}$, that is, terms containing only fields related to the high-energy bands are excluded. Therefore, after the Legendre transformation, $\mathcal{H}_{0}=-\sum_{i,\eta_{i}}\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{\tau}\eta_{i})}\partial_{\tau}\eta_{i}+\mathcal{L}_{0}$, where $\eta_{i}=\alpha_{i},\alpha_{i}^{\frac{1}{2}}$, and $e_{i\uparrow}$ (fields related to the low-energy bands), with $\frac{\partial\mathcal{L}_{0}}{\partial(\partial_{\tau}\eta_{i})}=\eta_{i}^{\dagger}$, the Lagrangian associated with $\mathcal{H}_{0}$ (up to order $J$) is given by

[TABLE]

We shall now focus on the $U\gg t$ perturbative expansion of $\mathcal{L}_{n}$, Eq. (22), which amounts to consider the most significant low-energy processes, after the use of Eqs. (36) and (37) for $d_{i\sigma}$ and $b_{i\sigma}$ in terms of the spinless Grassmann fields. However, terms allowing interband transitions between low- and high-energy bands do exist in $\mathcal{L}_{n}$. In this context, we apply a suitable second-order Rayleigh-Schrödinger perturbation theory Weng et al. (1991); Oliveira et al. (2009), consistent with the strong-coupling expansion, so that the modes associated with the high-energy bands are eliminated. Lastly, by adding $\mathcal{L}_{0}$ to the perturbative expansion of $\mathcal{L}_{n}$, which leads to the cancellation of the exchange terms in Eq. (40), the effective low-energy Lagrangian density of the $AB_{2}$ Hubbard model in the strong-coupling limit (up to order $J$) reads:

[TABLE]

where

[TABLE]

where

[TABLE]

in which case we took advantage of the symmetry of the $AB_{2}$ chain under the exchange operation $B_{1}\leftrightarrow B_{2}$, in correspondence with Eq. (25). From the above equations, we see that the kinetic term is represented by $\mathcal{L}^{(I)}$ and is related to the charge degrees of freedom only, whereas $\mathcal{L}^{(II)}$ describes the dynamics of the spin degrees of freedom coupled to the charge fields. On the other hand, $\mathcal{L}^{(III)}$ exhibit first-neighbor hopping contributions between charge degrees of freedom in the presence of $SU(2)$ gauge fields, while $\mathcal{L}^{(IV)}$ is the spin exchange term in the presence of the charge Grassmann fields.

IV half-filling regime

Let us now discuss some basic aspects of the localized magnetic properties related to the spin degrees of freedom. At half filling, i.e., $\delta=0$, we have $\langle\alpha_{i}^{\dagger}\alpha_{i}\rangle=1$, $\langle\alpha_{i}^{\left(1/2\right)\dagger}\alpha_{i}^{(1/2)}\rangle=1$, $\langle e_{i\uparrow}^{\dagger}e_{i\uparrow}\rangle=1$, and $\langle\alpha_{i}^{\dagger}e_{i\uparrow}\rangle=0$ (no band hybridization) as the electrons tend to fill up the lower-energy bands, whereas the higher-energy ones remain empty. As a consequence, a ferrimagnetic configuration of localized spins emerges, i.e., the charge degrees of freedom are completely frozen, such that $\langle\alpha_{i}^{\dagger}\partial_{\tau}\alpha_{i}\rangle=\langle\alpha_{i}^{\left(1/2\right)\dagger}\partial_{\tau}\alpha_{i}^{(1/2)}\rangle=\langle e_{i\uparrow}^{\dagger}\partial_{\tau}e_{i\uparrow}\rangle=0$, with forbidden hopping. Therefore, only terms from $\mathcal{L}^{II}$ and $\mathcal{L}^{IV}$ in Eqs. (42b) and (42d), respectively, give nonzero contributions and the resulting effective strong-coupling Lagrangian at half filling, defined in Eq. (41), reads:

[TABLE]

where the staggered factor $p_{i\alpha}$ was defined in Eq. (11), and use was made of the matrix transformations defined in Eq. (43) in order to sum up the squares of the $SU(2)$ gauge field products in the exchange contribution from $\mathcal{L}^{IV}$ in Eq. (42d). Now, using the following Legendre transform: $\mathcal{H}_{eff}^{J}=-\sum_{i\alpha\sigma}\frac{\partial\mathcal{L}_{eff}^{J}}{\partial(\partial_{\tau}U_{i\alpha})_{\sigma,\sigma}}(\partial_{\tau}U_{i\alpha})_{\sigma,\sigma}+\mathcal{L}_{eff}^{J},$ where $\frac{\partial\mathcal{L}_{eff}^{J}}{\partial(\partial_{\tau}U_{i\alpha})_{\sigma,\sigma}}=\theta(p_{i\alpha}\sigma)(U_{i\alpha}^{\dagger})_{\sigma,\sigma},$ we get the respective quantum Heisenberg Hamiltonian written in terms of the $SU(2)$ gauge fields at half filling as

[TABLE]

Further, using the definition of the $SU(2)/U(1)$ unitary rotation matrix Eq. (14), it is possible to write Weng et al. (1991, 1992); Oliveira et al. (2009); Ribeiro and Coutinho-Filho (2015) $\left|(U_{i\alpha}^{\dagger}U_{j\beta})_{\sigma,\sigma}\right|^{2}=\frac{1}{2}(1+\mathbf{n}_{i\alpha}\cdot\mathbf{n}_{j\beta}),$ where $\mathbf{n}_{i\alpha}=\sin(\theta_{i\alpha})\left[\cos(\phi_{i\alpha})\hat{\mathbf{x}}+\sin(\phi_{i\alpha})\hat{\mathbf{y}}\right]+\cos(\theta_{i\alpha})\hat{\mathbf{z}}$ is the unit vector pointing along the local spin direction. Lastly, by using the constraint as given in Eq. (19), we can identify the spin field $\{\mathbf{S}_{i\alpha}\}$ at the single occupied sites:

[TABLE]

where $p_{i\alpha}=+1\,\,(-1)$ at sites $\alpha=B_{1},\,B_{2}$ $(A)$, in order to obtain

[TABLE]

The above expression is indeed that of the quantum antiferromagnetic Heisenberg spin-1/2 model on the $AB_{2}$ chain in zero-field, which takes into account the effects of zero-point quantum spin fluctuations. In fact, to achieve this goal, we analyze the Hamiltonian, Eq. (47), by means of the spin-wave theory, which has proved very successful in describing the properties of the GS and low-lying excited states of spin models. The predicted results provide a check of the consistency of our approach and will be fully used in our description of the doped regime.

We shall first introduce boson creation and annihilation operators via the Holstein-Primakoff Fradkin (2013) transformation:

[TABLE]

for a down-spin on the $A$ site, and

[TABLE]

for an up-spin on the $B_{l}$ site, with $l=1,2$, and

[TABLE]

where $S$ is the spin magnitude, and $n_{r}=a_{i}^{\dagger}a_{i}$ or $b_{li}^{\dagger}b_{li}$. The operators $a_{i}^{\dagger}$ and $a_{i}$ (or $b_{li}^{\dagger}$, $b_{li}$ ) satisfy the boson commutation rules. Under the above transformation, the spin Hamiltonian, Eq. (47) is mapped onto the boson Hamiltonian:

[TABLE]

where

[TABLE]

is the classical GS energy and $\mathcal{H}_{1}$ and $\mathcal{H}_{2}$ are the quadratic and quartic (interacting) terms of the boson Hamiltonian, suitable to describe the quantum $AB_{2}$ Heisenberg model via a perturbative series expansion in powers of $1/S$. By Fourier transforming the boson operators, we find

[TABLE]

where we have defined the lattice structure factor as

[TABLE]

with $z$ denoting the coordination number ($z=4$ for the $AB_{2}$ chain), while $\rho=\pm 1/2$ connects the nearest neighbors $A\textrm{-}B_{1}$ and $A\textrm{-}B_{2}$ linked sites of sublattices $A$ and $B$, and

[TABLE]

For simplicity, we use the convention $1$ for $k_{1}$, $2$ for $k_{2}$, and so on. Also, the $\delta_{12,34}=\delta(k_{1}+k_{2}-k_{3}-k_{4})$ is the Kronecker $\delta$ function, and expresses the conservation of momentum to within a reciprocal-lattice vector $G$.

We shall consider $\mathcal{H}_{1}$ first, which is the term leading to linear spin-wave theory (LSWT). In fact, $\mathcal{H}_{1}$ is diagonalized using the following Bogoliubov transformation:

[TABLE]

[TABLE]

where $u_{k}$ and $v_{k}$ satisfy the constraint $u_{k}^{2}-v_{k}^{2}=1$. Thus,

[TABLE]

[TABLE]

[TABLE]

where $E_{1}$ is the $\mathcal{O}(S^{1})$ quantum correction to the GS energy, and $\epsilon_{k}^{0(\alpha,\beta)}$, $\epsilon_{k}^{0(\xi)}$ are the three spin-wave branches provided by LSWT, both in agreement with previous results Vitoriano et al. (2002); Yamamoto and Ohara (2007). In fact, it is well known that systems with a ferrimagnetic GS naturally have ferromagnetic and antiferromagnetic spin-wave modes as their elementary magnetic excitations (magnons). For the $AB_{2}$ chain, there are three spin-wave branches: an antiferromagnetic mode ($\epsilon_{k}^{0(\beta)}$) and two ferromagnetic ones ($\epsilon_{k}^{0(\alpha)}$ and $\epsilon_{k}^{0(\xi)}$). The mode $\epsilon_{k}^{0(\alpha)}$ is gapless at $k=0$, i.e., the Goldstone mode, with a quadratic (ferromagnetic) dispersion relation $\epsilon_{k}^{0(\alpha)}\sim k^{2}$. The other two modes are gapped. Notice that the gapped ferromagnetic mode $\epsilon_{k}^{0(\xi)}$ is flat, and is closely associated with ferrimagnetic properties at half filling Macêdo et al. (1995); Tasaki (1998b). Since the dispersive modes preserve the local triplet bond, they are identical to those found in the spin-$1/2$ - spin-1 chains Pati et al. (1997); Yamamoto et al. (1998); Ivanov (2000); Yamamoto (2004). These chains also exhibit interesting field-induced Luttinger liquid behavior da Silva and Montenegro-Filho (2017).

Now, our aim is to obtain the leading corrections to LSWT, i.e., second-order spin-wave theory to the GS energy, sublattice magnetizations and Lieb GS total spin per unit cell. In doing so, we develop a perturbative scheme for the description of this quartic term. First, we decompose the two-body terms by means of the Wick theorem, via normal-ordering protocol for boson operators. Conservation of momentum to within a reciprocal-lattice vector, implies: $k_{1}=k+q$, $k_{2}=p-q$, $k_{3}=k$ and $k_{4}=p$. Then, we need to look at the possible pairings of the 4 operators, as for example, in the first term of Eq. (55):

[TABLE]

Under this procedure, and by substituting the Bogoliubov transformation, Eqs. (56)-(57), into Eq. (55), we find

[TABLE]

where

[TABLE]

and the corresponding corrections for the spin-wave dispersion relations read:

[TABLE]

$\delta\epsilon_{k}^{(\beta)}$ is obtained from $\delta\epsilon_{k}^{(\alpha)}$ through the exchange of $u_{k}\leftrightarrow v_{k}$, and

[TABLE]

In Eqs. (62)-(64) above, the quantities $q_{1}$ and $q_{2}$ are defined by (thermodynamic limit)

[TABLE]

We remark that in deriving Eqs. (62)-(64), we have neglected terms containing anomalous products, such as, $\alpha_{k}^{\dagger}\beta_{k}^{\dagger}$ and vertex corrections.

Lastly, the above results of our perturbative $1/S$ series expansion lead to the effective Hamiltonian:

[TABLE]

where

[TABLE]

which can be read from Eqs. (52), (59), and (62), respectively, is the second-order result up to $\mathcal{O}(1/S)$ for the GS energy, and

[TABLE]

are the corresponding second-order spin-wave modes, where the linear and the second-order correction terms are given by Eq. (60) and Eqs. (63)-(65), respectively.

IV.1 Second-order spin-wave analysis

Our perturbative $1/S$ series expansion approach is able to improve the LSWT result for the gap $\Delta=J$ of the antiferromagnetic mode, which should be compared with the second-order result derived from $\epsilon_{k}^{(\beta)}$, Eqs. (60), (63) and (68), at $k=0$: $\Delta=(1+\sqrt{2}q_{2})J\simeq 1.676J$, in full agreement with similar spin-wave calculations for $AB_{2}$ Yamamoto and Ohara (2007) and spin-$1/2$-spin-$1$ Ivanov (2000); Yamamoto (2004) chains, and in agreement with numerical estimates using exact diagonalization, $\Delta=1.759J$, for both $AB_{2}$ Montenegro-Filho and Coutinho-Filho (2005) and spin-$1/2$-spin-$1$ Yamamoto et al. (1998) chains. On the other hand, the LSWT predicts a gap $\Delta_{flat}=J$ for the flat ferromagnetic mode ($\epsilon_{k}^{(\xi)}$) in $AB_{2}$ chain, whereas our second-order spin-wave theory finds, using Eqs. (60), (64) and (68): $\Delta_{flat}=(1-2q_{1}+\sqrt{2}q_{2})J\simeq 1.066J$, in full agreement with a similar spin-wave procedure Yamamoto and Ohara (2007). Surprisingly, the estimated value from Exact Diagonalization (ED) Montenegro-Filho and Coutinho-Filho (2005): $\Delta_{flat}=1.0004J$, lies between these two theoretical values. In fact, analytical approaches are still unable to reproduce the observed level crossing found in numerical calculations Montenegro-Filho and Coutinho-Filho (2005); Yamamoto and Ohara (2007) for the two ferromagnetic modes. This is probably due to the fact that the different symmetries exhibited by the localized excitation (flat mode) and the ferromagnetic dispersive mode are not explicitly manifested in the analytical approaches, so the levels avoid the crossing.

IV.2 Ground state energy

In the thermodynamic limit, the second-order result for the GS energy of the $AB_{2}$ chain per unit cell reads:

[TABLE]

We remark that, at half filling, we shall not consider the constant term $-JN_{c}$ in Eq. (51), with the purpose of comparison with preceding results. Performing the integration over the first BZ and taking $S=1/2$, we obtain that the GS energy per site at zero-field is given by $-0.4869J$. This result agrees very well with values obtained using exact diagonalization Takano et al. (1996) ($-0.485J$) and DMRG Niggemann et al. (1997) ($-0.4847J$) techniques. For the spin-$1/2$ - spin-1 chain, the value obtained using DMRG Pati et al. (1997) is $-0.72704J$. To compare it with our finding, we need to multiply this value by $2/3$ (ratio between the number of sites of the two chains), yielding $-0.48469J$.

IV.3 Sublattice magnetizations and Lieb GS total spin per unit cell

In order to derive results beyond LSWT, we introduce staggered magnetic fields coupled to spins $S_{i}^{A,z}$ and $S_{i}^{B_{l},z}$, with $l=1,2$, through the Zeeman terms: $-h_{A}\sum_{i}S_{i}^{A,z}$ and $-h_{B_{l}}\sum_{i}S_{i}^{B_{l},z}$, which are added to $\mathcal{H}_{eff}^{J}$ in Eq. (47). Thus, $\left\langle\right.S^{A,z}\left.\right\rangle$ and $\left\langle\right.S^{B_{l},z}\left.\right\rangle$ corresponding to sublattices $A$ and $B_{l}$ are obtained from $\left\langle\right.S^{A,z}\left.\right\rangle=-(1/N_{c})\sum_{i=1,2}[\partial E_{i}(h_{A})/\partial h_{A}]|_{h_{A}=0}$, and an analogous equation for $\left\langle\right.S^{B_{l},z}\left.\right\rangle$ using Eqs. (59) and (62):

[TABLE]

Carrying out the above integration, we obtain $\left\langle\right.S^{A,z}\left.\right\rangle=-0.316343$ and $\left\langle\right.S^{B_{l},z}\left.\right\rangle=0.408172$. These results are in good agreement with those obtained using DMRG Sierra et al. (1999) and ED Montenegro-Filho and Coutinho-Filho (2005) techniques: $\left\langle\right.S^{A,z}\left.\right\rangle=-0.2925$ and $\left\langle\right.S^{B_{l},z}\left.\right\rangle=0.3962$, respectively, and with values for $\left\langle\right.S^{A,z}\left.\right\rangle$ and $2\left\langle\right.S^{B_{l},z}\left.\right\rangle$ for the spin-$1/2$ - spin-1 chain Pati et al. (1997); Yamamoto et al. (1998); Ivanov (2000); Yamamoto (2004). Although at zero temperature, the sublattice magnetizations are strongly reduced by quantum fluctuations, as compared with their classical values, the unit cell magnetization remains $S_{L}\equiv 1/2$, where $S_{L}$ is the Lieb GS total spin per unit cell, in full agreement with Lieb’s theorem Lieb (1989); Montenegro-Filho and Coutinho-Filho (2005) for bipartite lattices:

[TABLE]

with $N_{A}(N_{B})$ denoting the total number of spins in sublattice $A(B)$ per unit cell.

V $\mathbf{t\textrm{-}J}$ Hamiltonian: Doping-Induced Phases, Ground state

Energy and Total Spin

In this section, we shall derive the corresponding $t$-$J$ Hamiltonian suitable to describe the strongly correlated $AB_{2}$ Hubbard chain in the doped regime, in which case both charge (Grassmann fields) and spin [$SU(2)$ gauge fields] quantum fluctuations are considered on an equal footing. Indeed, the $t$-$J$ Hamiltonian can be derived by means of the following Legendre transformation to Eq. (41):

[TABLE]

where ${\frac{\partial\mathcal{L}_{eff}}{\partial(\partial_{\tau}\nu_{i})}=\nu_{i}^{\dagger}}$ with ${\nu_{i}=\alpha_{i},\alpha_{i}^{\frac{1}{2}},e_{i\uparrow}}$; $\frac{\partial\mathcal{L}_{eff}}{\partial(\partial_{\tau}U_{i}^{(b)})_{\sigma,\sigma}}=\theta(-\sigma)(U_{i}^{(b)\dagger})_{\sigma,\sigma}\alpha_{i}^{(\frac{1}{2})\dagger}\alpha_{i}^{(\frac{1}{2})}$, and $\frac{\partial\mathcal{L}_{eff}}{\partial(\partial_{\tau}U_{i}^{(d,e)})_{\sigma,\sigma}}=\theta(\sigma)\frac{1}{2}[(U_{i}^{(d,e)\dagger})_{\sigma,\sigma}(\alpha_{i}^{\dagger}\alpha_{i}+e_{i\uparrow}^{\dagger}e_{i\uparrow})+(U_{i}^{(e,d)\dagger})_{\sigma,\sigma}(\alpha_{i}^{\dagger}e_{i\uparrow}+e_{i\uparrow}^{\dagger}\alpha_{i})]$, from which we can write the effective $t$-$J$ Hamiltonian as

[TABLE]

where

[TABLE]

and

[TABLE]

Notice that Eqs. (74) and (75) are identical to Eqs. (42c) and (42d), since Eqs. (42a) and (42b) were eliminated through the Legendre transformation.

Some digression on $\mathcal{H}_{eff}^{t\textrm{-}J}$ is in order. One of the key properties of quasi-1D interacting quantum systems is the phenomenon of spin-charge separation, leading to the formation of spin and charge-density waves, which move independently and with different velocities. It has been demonstrated Montenegro-Filho and Coutinho-Filho (2006) that for $\delta>2/3$ the low-energy physics of the doped $AB_{2}$ Hubbard chain in the $U=\infty$ coupling limit is described in terms of the Luttinger-liquid model, with the spin and charge degrees of freedom decoupled. Most importantly, it has been shown that for the $AB_{2}$ $t$-$J$ Hubbard chains Montenegro-Filho and Coutinho-Filho (2014) charge and spin quantum fluctuations are practically decoupled, as suggested by the emergence of charge-density waves in anti-phase with the modulation of the ferrimagnetic order. One can make use of this feature to formally split each term of the $t$-$J$ Hamiltonian, Eq. (73)-(75) , into a product of two independent terms acting on different Hilbert spaces, i.e., we can enforce spin-charge separation and calculate the charge and spin correlation functions in a decoupled fashion.

Therefore, from the above discussion, we shall consider that the charge correlation functions are well described by an effective spinless tight-binding model Montenegro-Filho and Coutinho-Filho (2006); Oliveira et al. (2009); Vitoriano and Coutinho-Filho (2010), since the hole (charge) density waves develop along the $x$-axis and in anti-phase with the modulation of the ferrimagnetic structure, as numerically observed in Fig. 2(b) of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]. So, using Eqs. (33), with $a/2\rightarrow a$ (effective lattice spacing of the linear chain: distance between $A$ and $B$ sites, see Fig. 2(a) of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]), we find

[TABLE]

with $\left.|\varPsi_{0}\right\rangle$ being the hole-doped ferrimagnetic GS, where ${k_{F}(\delta)=\pi\frac{N_{h}}{N}\equiv\pi\delta}$ is the Fermi wave vector of the spinless tight-binding holes. In the same fashion: ${\left\langle\right.\alpha_{i}^{\dagger}\alpha_{i+1}^{\left(1/2\right)}\left.\right\rangle=\frac{2}{\pi}\sin[k_{F}(\delta)]}$ and ${\left\langle\right.\alpha_{i}^{\left(1/2\right)\dagger}e_{i\uparrow}\left.\right\rangle=\left\langle\right.e_{i\uparrow}^{\dagger}\alpha_{i+1}^{\left(1/2\right)}\left.\right\rangle=0}$; while $\left\langle\right.\alpha_{i}^{\dagger}\alpha_{i}\left.\right\rangle=\left\langle\right.\alpha_{i}^{\left(1/2\right)\dagger}\alpha_{i}^{\left(1/2\right)}\left.\right\rangle=\left\langle\right.e_{i\uparrow}^{\dagger}e_{i\uparrow}\left.\right\rangle=(1-\frac{1}{2\pi}\int_{-k_{F}(\delta)}^{k_{F}(\delta)}dk)=(1-\delta)$. Here, we remark that the itinerant holes away from half filling are associated with the lower-energy dispersive $\alpha_{k}$ and $\alpha_{k}^{\left(1/2\right)}$ bands [see Fig. 1LABEL:sub@Fig_1b in Sec. (II)], thus contributing to the kinetic Hamiltonian in Eq. (74). On the other hand, the local correlations related to the lower-energy bands $\alpha_{k}$, $\alpha_{k}^{\left(1/2\right)}$, and $e_{k\uparrow}$, contribute equally to the exchange Hamiltonian in Eq. (75). Thereby, using the above tight-binding results for the charge correlation functions, $\mathcal{H}_{eff}^{t\textrm{-}J}$ in Eqs. (73)-(75) gives rise to the $\delta$-dependent Hamiltonian, $\mathcal{H}_{eff}^{t\textrm{-}J}(\delta)=\mathcal{H}_{eff}^{t}(\delta)+\mathcal{H}_{eff}^{J}(\delta)$, written below:

[TABLE]

where the sum over $\sigma$ was evaluated in Eq. (74) and the square of the $SU(2)$ gauge field products in the exchange contribution have been summed up in Eq. (75), so that this contribution is just $(1-\delta)$ times $\mathcal{H}_{eff}^{J}$ at half filling, Eq. (45), or alternatively, in terms of spin fields, Eq. (47), or spin-waves, Eqs. (66)-(68). On the other hand, the $SU(2)$ gauge fields matrix elements: $\left(U_{i}^{(b)\dagger}U_{i}^{(d)}\right)_{\downarrow\uparrow}$ and $\left(U_{i}^{(d)\dagger}U_{i+1}^{(b)}\right)_{\uparrow\downarrow}$, that appear in the kinetic contribution of Eq. (77), can be written in terms of the spin fields Oliveira et al. (2009); Ribeiro and Coutinho-Filho (2015) as

[TABLE]

and $\left(U_{i}^{(d)\dagger}U_{i+1}^{(b)}\right)_{\uparrow\downarrow}$ is obtained from Eq. (78) through the replacement $S_{i}^{A,z}\rightarrow S_{i+1}^{A,z}$, in which case we took advantage of the $U(1)$ gauge freedom and Eq. (46). Notice that these square-root matrix elements depend on $z$-spin components only.

At this stage, it will prove useful, in the calculation of the GS total spin in the doped regime, to consider $\mathcal{H}_{eff}^{t\textrm{-}J}(\delta,h)$ which describes the system in the presence of a homogeneous magnetic field $\mathbf{h}=h\mathbf{\hat{z}}=(-h_{A}+h_{B_{1}}+h_{B_{2}})\mathbf{\hat{z}}$, where the staggered fields point along the local corresponding magnetizations in the ferrimagnetic phase have the same magnitude $h$. The magnetic field couple with the spin fields through the Zeeman term (see Sec. IV.3) and with the charge degrees of freedom through the magnetic orbital coupling in the Landau gauge: $\mathbf{A}=hx\mathbf{\hat{y}}$. Since our aim is to study doping effect on the magnetization, we shall assume vanishingly small magnetic field in the context of linear response theory and perturbative expansion in the strong-coupling regime. Additionally, the magnetic orbital coupling can be considered through the so-called Peierls substitution Essler et al. (2005); Vidal et al. (2000); *Gulacsi2007: $t\rightarrow te^{i\int_{{}_{i\alpha}}^{{}_{j\beta}}\mathbf{A}\cdot d\mathbf{l}}$, where $i\alpha$ and $j\beta$ are first-neighbor sites, and the flux quantum $\phi_{0}=hc/e\equiv 1$. If one consider that the carrier is at the site $iA$, we have four hopping possibilities: $iA\rightarrow iB_{1,2}$ and $iA\rightarrow(i+1)B_{1,2}$, so the total phase $\phi$ acquired by the carrier in this prescription satisfies Stokes’ theorem: $\phi=\ointctrclockwise_{\mathrm{unit\,cell}}\mathbf{A}\cdot d\mathbf{l}=\iint_{S}\mathbf{h}\cdot d\mathbf{S}=ha^{2}$ $(a\equiv 1)$. We also remark that, in order to obtain real values for the zero-field staggered magnetizations, we have considered, for convenience, an imaginary gauge transformation Ribeiro and Coutinho-Filho (2015); Hatano and Nelson (1996): $\mathbf{A}\rightarrow i\mathbf{A}$. Therefore, by placing Eq. (78) and the similar matrix element into the kinetic term in Eq. (77), making the above Peierls substitution, and using the Holstein-Primakoff and Bogoliubov transformations introduced in Eqs. (48)-(50) and Eqs. (56)-(57), respectively, up to order $\mathcal{O}(S^{-1})$, we arrive at the following diagonalized kinetic Hamiltonian $\mathcal{H}_{eff}^{t}(\delta,h)$:

[TABLE]

where the doped-induced contributions for the spin dispersion relations are evidenced in the last three terms. On the other hand, by adding the Zeeman terms (see Sec. IV.3) to the exchange contribution $\mathcal{H}_{eff}^{J}(\delta)$, given in Eq. (77), we obtain $\mathcal{H}_{eff}^{J}(\delta,h)$. Lastly, by adding the kinetic and the exchange contributions, we arrive at the effective $t$-$J$ Hamiltonian in the presence of a magnetic field:

[TABLE]

where $E_{GS}^{J}$ is given by Eq. (67) and (69), and the corresponding spin-wave modes [see Eqs. (79), (60), (63)-(65), and (68)] of the doped $AB_{2}$ $t$-$J$ chain read:

[TABLE]

$\epsilon_{k}^{(\beta)}(\delta)$ is obtained from $\epsilon_{k}^{(\alpha)}(\delta)$ through the exchange $u_{k}\leftrightarrow v_{k}$ and the replacement $\alpha\rightarrow\beta$, while

[TABLE]

We find it instructive to comment on the analytical structure of the above equations. Firstly, we mention the presence of the Bogoliubov parameters [see Eqs. (57)] in a symmetric form in the kinetic terms of Eq. (81) and its analogous for $\epsilon_{k}^{(\beta)}(\delta)$; besides, although the flat mode is strongly affected by the presence of holes, it remains dispersionless. In addition, using Eqs. (80) and (65), the total GS energy (no spin-wave excitations) per unit cell in the thermodynamic limit is readily obtained:

[TABLE]

where $\left\langle\right.S^{A,z}\left.\right\rangle$ and $\left\langle\right.S^{B_{l},z}\left.\right\rangle$ are the calculated sublattice magnetizations, at half filling and zero-field, given by Eqs. (70).

In subsections V.1, V.2, and V.3, we will show that the underlying competing physical mechanisms: the magnetic orbital response and the Zeeman contribution embedded in Eqs. (80)-(83) will dramatically affect the behavior of the system under hole doping and, in particular, will lead to spiral IC spin structures, the breakdown of the spiral ferrimagnetic GS at a critical value of the hole doping, a region of phase separation, and RVB states at $\delta\approx 1/3$.

V.1 Doped regime: Spin-wave modes

Before we go one step further to discuss relevant macroscopic quantities, i.e., the GS energy and total spin in the doped regime, we shall first undertake a detailed study, at a microscopic level, of the hole-doping effect on the calculated spin-wave branches given by Eqs. (81)-(82).

Fig. 2 depicts the second-order spin-wave dispersion relations at $J/t=0.3$ and for the indicated values of $\delta$. Without loss of generality, we set $t=1$ in our numerical computations. At half filling, the antiferromagnetic mode $\epsilon_{k}^{(\beta)}$, together with the two ferromagnetic modes: the dispersive $\epsilon_{k}^{(\alpha)}$ and the flat one $\epsilon_{k}^{(\xi)}$, are shown in Fig. 2LABEL:sub@Fig_2a, which are defined in Eq. (68), and can be plotted using Eqs. (60) and (63)-(65).

As the hole doping increases slightly, the abrupt decrease of the peaks at $k=0$ and $k=\pi$ of the numerical DRMG structure factor (see Fig. 3 of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]), associated with the ferrimagnetic order, manifests itself here through the opening of a gap in the ferromagnetic Goldstone mode $\epsilon_{k}^{(\alpha)}$, as seen in Fig. 2LABEL:sub@Fig_2b, thus indicating that the system loses its long-range order. Note that the antiferromagnetic mode $\epsilon_{k}^{(\beta)}$ is also similarly shifted. On the other hand, although the dispersion relation is modified for small values of the wave vector $k$, the minimum value of $\epsilon_{k}^{(\alpha)}$ still remains at $k=0$ up to the onset of the formation of spiral IC spin structures at $\delta_{c}(\mathrm{IC})=0.043$ (a value that should be compared with the numerical DMRG estimate of $\delta\approx 0.055\pm 0.012$), characterized by the flattening of the dispersive spin-wave branches around zero. Upon further increase of $\delta$, two minima form (around $k=0$) and move away from each other as one enhances the hole doping. This behavior is the signature of the occurrence of spiral IC spin structures (see Fig. 3 of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]).

Fig. 2LABEL:sub@Fig_2c shows the onset of phase separation (PS) at $\delta(\mathrm{PS})=0.165$ for $J=0.3$, which is characterized by the overlap of the two ferromagnetic modes at $k=0$. The signature of this regime is the spatial coexistence of two phases: spiral IC spin structures at $\delta(\mathrm{PS})=0.165$ and RVB states at $\delta\approx 1/3$, in very good agreement with the numerical estimate of $\delta_{\mathrm{IC-PS}}\approx 0.16$ Montenegro-Filho and Coutinho-Filho (2014). At $\delta\approx 1/3$, the flat mode has the lowest energy, as illustrated in Fig. 2LABEL:sub@Fig_2d. This behavior indicates that the RVB state is the stable phase at $\delta\approx 1/3$ and $J=0.3$ Montenegro-Filho and Coutinho-Filho (2014), and also in agreement with the numerical DMRG studies Sierra et al. (1999); Montenegro-Filho and Coutinho-Filho (2006) and analytical prediction at $U=\infty$ Oliveira et al. (2009).

In order to better understand the rich variety of doping-induced phases in the system, in Fig. 3 we plot the evolution of the wave vector $k_{\mathrm{min}}$ corresponding to the local minimum of $\epsilon_{k}^{(\alpha)}(\delta)$, upon increasing the hole doping $\delta$ from [math] to $1/3$. The wave vector $k_{\mathrm{min}}$ remains zero until it hits the onset doping value $\delta_{c}(\mathrm{IC})=0.043$, beyond which a square-root growth behavior takes place Chakrabarty et al. (2012): $[\delta-\delta_{c}(\mathrm{IC})]^{1/2}$ (blue line), for $\delta$ close to $\delta_{c}(\mathrm{IC})$. The square-root growth behavior is the signature of the occurrence of a second-order quantum phase transition from the doped ferrimagnetic phase to the IC spiral ferrimagnetic state with a non-zero value of the total GS spin, $S_{GS}$. This result is supported by the behavior of $\Delta_{k}\equiv k_{\mathrm{max}}-\pi$ at which the local maximum of the numeric DMRG structure factor $S(k)$ near $k=\pi$ is observed, as shown in the inset of Fig. 3 (taken from the inset of Fig. 3(b) of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]). For further increase of hole doping our result deviates from the square-root growth behavior and some very interesting features are to be noticed. The value of $\delta_{c}=0.08$ indicates the breakdown of the total $S_{GS}$ in the IC phase, as will be confirmed by the explicit calculation of $S_{GS}$, a macroscopic quantity, in Section V.3. Thus, for $0.08<\delta<0.165$ the system displays an IC phase with zero $S_{GS}$, in agreement with the DMRG data (see Fig. 1(c) of Ref. [Montenegro-Filho and Coutinho-Filho, 2014]). At $\delta(\mathrm{PS})=0.165$ the system exhibits a first-order transition accompanied by the spatial phase separation regime: the IC phase with zero $S_{GS}$ and modulation fixed by $\delta(\mathrm{PS})$ in coexistence with the short-range RVB states at $\delta\approx 1/3$, also consistent with the DMRG data plotted in Fig. 4 of Ref. [Montenegro-Filho and Coutinho-Filho, 2014].

Lastly, we emphasize that, despite the occurrence of several doping-induced phases in the DMRG studies Montenegro-Filho and Coutinho-Filho (2014): Lieb ferrimagnetism, spiral IC spin structures, RVB states with finite spin gap, phase separation, and Luttinger-liquid behavior, it is surprising and very interesting that the second-order spin-wave modes remain stable up to $\delta\approx 1/3$, with predictions in very good agreement with the DMRG studies Montenegro-Filho and Coutinho-Filho (2014). In this context, it is worth mentioning the long time studied case of rare earth metals Cooper and Elliott (1963); *[Seealso:][; Chapter1; p.187.]Elliott1972magnetic, where an external magnetic field can induce non-trivial phase transitions involving spiral spin structures, well described by spin-wave theory.

V.2 Doped regime: Ground state energy

Performing the integration over the first BZ in Eqs. (65) and (69) and setting $S=1/2$ in Eq. (83), we find that the $AB_{2}$ $t$-$J$ ground state energy per unit cell as a function of hole doping in zero-field reads:

[TABLE]

We shall now examine the case of small hole doping away from half filling, i.e., with hole concentration ranging from $\delta=0$ up to $\delta=0.2$ for two values of $J$: $0.1$ and $0.3$.

In Fig. 4, we show the evolution of the GS energy per unit cell of the $AB_{2}$ $t$-$J$ model as a function of hole doping for both mentioned values of $J$, and the comparison was made with the numerical DMRG data Montenegro-Filho and Coutinho-Filho (2014). From the two results at $J=0.3$, the only quantitative difference induced by the increase of the hole concentration is a crossing feature around $\delta\approx 0.1$, where our analytical result slightly change its behavior by lowering the energy with respect to the numerical data Montenegro-Filho and Coutinho-Filho (2014). In fact, because our model assumes a ferrimagnetic state as the starting point, this change of behavior suggests that we have entered in a region of strong magnetic instabilities, and possibly indicating a smooth transition to an incommensurate phase with zero GS total spin beyond $\delta\approx 0.1$, as confirmed by the numerical data in Ref. [Montenegro-Filho and Coutinho-Filho, 2014] and illustrated in Fig. 3. On the other hand, at $J=0.1$, although our results reproduce the numerical data with an acceptable agreement, we observe a discrepancy that increases with $\delta$. The cause of such discrepancy will be discussed in the next subsection.

With the purpose of determining the interplay between the contribution of magnetic exchange and the itinerant kinetic energy to the zero-field GS energy Eq. (83), we take $J=0.3$ and show its evolution with doping in Fig. 5. We can see in the insets, Fig. 5LABEL:sub@Fig_5a and Fig. 5LABEL:sub@Fig_2b, the competitive behavior of the two energetic contributions, i.e., the contribution of the exchange energy increases linearly with $\delta$, while a practically linear decrease of the hopping term is observed as one enhances the hole doping. This competition indicates that a phase transition to a paramagnetic phase should occur at some critical concentration value.

V.3 Doped regime: Ground state total spin

The existence of a transition from an IC spiral ferrimagnetic phase to an IC paramagnetic one is a most interesting feature observed numerically in doped $AB_{2}$ $t$-$J$ Hubbard chains Montenegro-Filho and Coutinho-Filho (2014). In order to firmly corroborate the mentioned transition, we have calculated the GS total spin per unit cell as a function of hole doping, $S_{GS}(\delta)=\sum_{\alpha}\,\left\langle\right.S^{\alpha,z}\left.\right\rangle(\delta)$, with $\alpha=A,B_{1},B_{2}$, by means of the zero-field derivative of Eq. (83):

[TABLE]

We thus find

[TABLE]

where $+$ ($-$) corresponds to sublattice $\alpha=A$ ($\alpha=B_{1,2}$), and $\left\langle\right.S^{A,z}\left.\right\rangle$ and $\left\langle\right.S^{B_{l},z}\left.\right\rangle$ are given by Eqs. (70). Therefore, by performing the integration over the first BZ of the three contributions in Eq. (86), we finally obtain:

[TABLE]

where $S_{L}=\sum_{\alpha}\,\left\langle\right.S^{\alpha,z}\left.\right\rangle=1/2$ is Lieb’s reference value for the GS total spin per unit cell at half filling and zero-field (see Section IV.3).

In Fig. 6 we plot the evolution of $S_{GS}$, normalized by $S_{L}$, as a function of $\delta$, and compare it with the numerical data from DMRG and Lanczos techniques Montenegro-Filho and Coutinho-Filho (2014), for $J=0.3$ (red squares) and $J=0.1$ (blue circles). In the latter (former) case, the system undergoes a transition from the modulated itinerant ferrimagnetic phase to an incommensurate phase with zero (nonzero) $S_{GS}$. Notice that, in both cases, the transition is characterized by a decrease of $S_{GS}$ from $S_{L}$ to [math] or to a residual value, regardless of the value that $S_{GS}$ takes after the transition. Indeed, at $J=0.1$ and $\delta>0.1$, the formation of magnetic polarons (onset of the Nagaoka phenomena that sets in as $U\rightarrow\infty$) with charge-density waves in phase with the modulation of the ferrimagnetic structure, as indicated by the DMRG data Montenegro-Filho and Coutinho-Filho (2014), leads to an incommensurate phase with nonzero $S_{GS}$.

Most importantly, we can observe in Fig. 6 that the value of $S_{GS}$ decreases practically linearly with $\delta$ until the magnetic order is completely suppressed at $\delta_{c}\approx 0.08$. This behavior is supported by numerical results Montenegro-Filho and Coutinho-Filho (2014), particularly in the regime where the Nagaoka phenomenon is not manifested, that is, at $J=0.3$, as indicated in Fig. 3. In this regime, spin and charge quantum fluctuations destabilize the ferrimagnetic structure and trigger a transition to an incommensurate paramagnetic phase at $\delta_{c}$, with $S_{GS}\sim(\delta-\delta_{c})\rightarrow 0$.

VI CONCLUSIONS

In summary, we have presented a detailed analytical study of the large-U Hubbard model on the quasi-one-dimensional $AB_{2}$ chain. We used a functional integral approach combined with a perturbative expansion in the strong-coupling regime that allowed us to properly analyze the referred system at and away from half filling.

At half filling, our model was mapped onto the quantum Heisenberg model, and analyzed through a spin-wave perturbative series expansion in powers of $1/S$. We have demonstrated that the GS energy, spin-wave modes, and sublattice magnetizations are in very good agreement with previous results.

In the challenging hole doping regime away from half filling, the corresponding $t\textrm{-}J(=4t^{2}/U)$ Hamiltonian was derived. Further, under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime has unveiled the occurrence of spatially modulated spin structures and the emergence of phase separation (first-order transition) in the presence of resonating-valence-bond states. The doping-dependent GS energy and total spin per unit cell are also calculated, in which case the collapse of the spiral magnetic order at a critical hole concentration was observed. Remarkably, our above-mentioned analytical results in the doped regime are in very good agreement with density matrix renormalization group studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the ferrimagnetic structure.

Finally, we stress that our reported results evidenced that the present approach, also used in a study on the compatibility between numerical and analytical outcomes of the large-U Hubbard model on the honeycomb lattice, was proved suitable for the $AB_{2}$ chain (a quasi-1D system), where the impact of charge and spin quantum fluctuations are expected to manifest in a stronger way. We thus conclude that our approach offers a quite powerful analytical description of hole-doping induced phases away from half filling in low-dimensional strongly-correlated electron systems.

Acknowledgements.

We appreciate interesting discussions with R. R. Montenegro-Filho. This work was supported by CNPq, CAPES and FACEPE/PRONEX (Brazilian agencies).